Uniqueness of the Kerr-de Sitter spacetime as an algebraically special solution in five dimensions

TL;DR

This paper classifies five-dimensional vacuum Einstein solutions with a cosmological constant that are algebraically special and finds they are essentially unique, corresponding to the Kerr-de Sitter spacetime, unlike the four-dimensional case.

Contribution

It proves the uniqueness of the five-dimensional Kerr-de Sitter solution among algebraically special vacuum solutions with a non-degenerate optical matrix.

Findings

Solution is specified by three parameters.

Locally isometric to Kerr-de Sitter or related solutions.

Contrasts with four-dimensional case where solutions are not unique.

Abstract

We determine the most general solution of the five-dimensional vacuum Einstein equation, allowing for a cosmological constant, with (i) a Weyl tensor that is type II or more special in the classification of Coley et al., (ii) a non-degenerate "optical matrix" encoding the expansion, rotation and shear of the aligned null direction. The solution is specified by three parameters. It is locally isometric to the 5d Kerr-de Sitter solution, or related to this solution by analytic continuation or taking a limit. This is in contrast with four dimensions, where there exist infinitely many solutions with properties (i) and (ii).